Wow.. Your corolla rocks then. I doubt my judgement, even though I own a city since 10 years and a corolla since 5, City at 220k kms on the ODO and the corolla with 180k... Difference of opinion is the beauty of this world... 
FYI... My City runs on 195/60/R15. I can't feel it going above 120kph whereas corolla shouts and tells everyone in the cabin when its pushed past 100 kph... On a normal road I mean... Not motorway.
Moreover, I can't figure out a way of how your car goes above 160. Mine was just 2k driven when I tried pushing it hard but it won't go above 160, the needle seemed stuck.
I gave up, because it wasn't possible driving it at those speeds for long, moreover an exhaust of adrenaline in my opinion since you know its not gonna stop if something pops in front of you in a corolla. Again, your machine rocks. Mine sucks I think.
And I have no idea of how a stiffer suspension may account for a worse handling and road grip...!
May be you have your own ways of looking at things and rest of the wheeler community has their own. Because all I have heard and seen with my senses in this world is a sports suspension means road grip and its ALWAYS stiffer... Not soft.
10th Gen Corolla is a big jhoola, I owned once, didn't like it, sold it. Just because I could rarely feel it responding to my commands from inside the cockpit, all you can feel is moving, owing to the fact that you are looking at ground moving under you with immense speed.
What you wonder is correct, but what I wonder can't be classified as wrong either: Why do we always look at people as if they're all the same?
No offense meant of course!
Just FYI, I own civics, a corolla, an idsi city, an ivtec city and whatever I post here is based upon MY OWN judgment and personal expertise with the machines over time. I don't like bragging a lot but still, you looked at my posts otherwise. 
P.S: No civic vs altis, No JDM vs PKDM. Both comparisons suck, aren't meant to be and have been discussed TOO many times to be interesting anymore.